BEIJING (Reuters) - U.S. President Donald Trump's warm words for Chinese counterpart Xi Jinping as a "good man" will reassure Beijing that he finally understands the importance of good ties, but risks leaving America's regional allies puzzling over where they fit into the new order.

The budding relationship between the two leaders appeared highly unlikely when Trump was lambasting China on the campaign trail for stealing U.S. jobs with unfair trade polices.

In December, after winning office, he upended protocol by taking a call from the president of self-ruled Taiwan, which China regards as its own territory.

A few months on, after meeting Xi at his Florida residence earlier in April, Trump appears to have done a complete volte-face, praising Xi for trying hard to rein in nuclear-armed North Korea and rebuffing Taiwan's president's suggestion of another call.

But the big question is whether the rapprochement will last. Trump also expressed admiration for Russian President Vladimir Putin during the 2016 presidential campaign, but that relationship has since chilled.

Chinese officials will no doubt be pleased by Trump's overtures, said Jia Qingguo, a leading academic who has advised the government on foreign policy.

"People will say that the only thing we know for sure about Donald Trump's administration is uncertainty and unpredictability," said Jia, dean of the School of International Studies at the elite Peking University.

"But judging from what he has been saying and doing, it's quite reassuring as far as China is concerned. Certainly I think people have developed more positive views about the Donald Trump administration here and we have a lot of expectations that we can work together constructively."

For China's neighbors, it is a little more complicated.

On one level, a healthy relationship between the world's two biggest economies suits everyone.

"It's hugely positive that there's been a reasonably constructive start to the bilateral dialogue between those two countries," Tom Lembong, Indonesia's investment chief and close aide to President Joko Widodo, told Reuters.

But long-time allies may also be wondering just how far Washington still has their back.

Shashank Joshi, senior fellow at the Royal United Services Institute in London, said countries such as Japan and South Korea could lose influence if Trump's focus on enlisting Xi's help over North Korea creates a "sort of U.S.-China G2".

"There are competing instincts within Trump pushing him in opposite directions," said Joshi.

"His nationalism pushes him towards competition with China, but his deal-making instinct, his openness to personal influence, and his affinity for strongmen pushes him towards Xi, especially if he can show results on North Korea."

But Trump, who has long touted his deal-making ability as a real estate developer, has also made clear his approach to China is transactional. He is so focused on securing cooperation against North Korea, his top national security priority, that he has even publicly promised to go easier on Beijing over critical trade issues in return.

Some of Trump's aides doubt, however, that China will do enough to restrain North Korea's nuclear and missile programmers. Some experts believe the thaw between the economic rivals could be fleeting if Xi fails to come through on the North Korean issue.

SOUTH CHINA SEA

Singapore-based security expert Ian Store said he believed Trump's remarks would be closely scrutinized by Southeast Asian leaders looking for signs of an emerging Asia strategy.

"Most would welcome a calm, co-operative relationship between China and the U.S., but they will be deeply concerned at anything that looks like Trump will give Xi a free hand over the South China Sea dispute, or elsewhere," said Storey, who is based at the ISEAS Yusof Ishak Institute.

The administration has so far sent out mixed rhetorical signals over the hotly disputed South China Sea. China's extensive claims to the vital global trade route are challenged by Brunei, Malaysia, the Philippines and Vietnam, as well as Taiwan.

The U.S. has increased naval deployments in the South China Sea in recent years amid roiling tensions and extensive island-building by China but, under Trump, its warships have yet to challenge China with a so-called freedom of navigation patrol close to disputed islets and reefs.

A Trump administration official has told Reuters the United States wants to avoid antagonizing China on sensitive issues like the South China Sea for now while waiting to see how far Beijing will go tightening the screws on North Korea. But the official, speaking on condition of anonymity, said this did not mean abandoning efforts to counter China's growing military and economic might in the Asia-Pacific region.

Admiral Harry Harris, the chief of the U.S. Pacific Command, told the U.S. Congress this week that he expected to be carrying out such patrols in the South China Sea soon, and repeated earlier concerns at China's continued militarization of the area.

"Given Trump's newfound friendship with Xi Jinping, it might make it significantly harder for the Pacific Command to get its plans approved for the next freedom of navigation patrols," Storey said.

UNPREDICTABLE

In Japan, often at odds with China over what Beijing views as Tokyo's failure to properly atone for World War Two, a Japanese government source sought to downplay any impact the burgeoning Trump-Xi friendship might have on Japan-U.S. ties.

"Trump's softened approach to Xi may seem to be some kind of shift in the balance of power but security cooperation between Japan and the United States is extremely stable and has been confirmed in the face of the current crisis situation in North Korea," the source told Reuters.

The tricky issue of Taiwan has not gone away either, and is one of several that could upset relations.

Democratic Taiwan has many friends in Washington who will not want to allow autocratic China to get its way with the island, and the United States is bound by law to provide Taiwan with the means to defend itself.

Wang Dong, associate professor of international studies at Peking University, said China would remain on alert for another change of direction by Trump.

"There are reasons for optimism, but we are still being realistic. There are still issues out there, from Taiwan to the South China Sea," he said.

One Beijing-based Western diplomat told Reuters that, while China might be pleased to see Trump hang ally South Korea out to dry with his criticism of their free trade deal and demand Seoul pay $1 billion to host a U.S. anti-missile system China has strongly opposed, China should not have any illusions.

"He's so unpredictable who knows what he'll say next week or next month?" said the diplomat, speaking on condition of anonymity. "His mood turns on a pin."

(Additional reporting by Michael Martina in Beijing; Linda Sieg in Toyko; Kanupriya Kapoor and Karen Lema in Manila; Sanjeev Miglani in New Delhi; Greg Torode in Hong Kong; Matt Spetalnick and David Brunnstrom in Washington; Editing by Alex Richardson and Frances Kerry)